ASN6 a,b- fixing

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Problem Statement

6(a) Show ${ℱ}[\underset{-\mathrm{\infty }}{\overset{t}{\int }}s(\lambda )d\lambda ]=\frac{S(f)}{j2\pi f}\text{ if }S(0)=0$. Hint: $S(0)=S(f){|}_{{f=0}_{}}=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}s(t)e^{-j2\pi (f\to 0)t}dt=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}s(t)dt$

6(b) If $S(0)\ne 0$ can you find ${ℱ}[\underset{-\mathrm{\infty }}{\overset{t}{\int }}s(\lambda )d\lambda ]$ in terms of $S(0)$?

Answer

a)

Remember dummy variable $\lambda =t-t_0$ Then $s(\lambda )=s(t-t_0)={ℱ}[S(f)-S(f_0)]$ and $\underset{-\mathrm{\infty }}{\overset{t}{\int }}s(\lambda )d\lambda =\underset{-\mathrm{\infty }}{\overset{t}{\int }}{ℱ}[S(f)-S(f_0)]d\lambda$

$f_0=0$ where $S(0)=S(f){|}_{f=0}=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}s(t)e^{-j2\pi ft}dt=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}s(t)dt$

$\text{ if }S(0)=0\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}s(t)dt=0$

$\underset{-\mathrm{\infty }}{\overset{t}{\int }}s(\lambda )d\lambda =\underset{-\mathrm{\infty }}{\overset{t}{\int }}{ℱ}[S(f)]dt$

Failed to parse (syntax error): {\displaystyle \mathcal{F}\left[ S (f)- S(f_0) \right] = \int_{- \infty}^{t} e^{-j2 \pi f t} \,dt \int_{- \infty}^{\infty} S(f)\,df = \frac{ e^{-j2 \pi f t}} {-j2 \pi f }\int_{- \infty}^{\infty} S(f) \,df =\infty}^{\infty} S(f)\frac{ e^{-j2 \pi f t}} {-j2 \pi f }\,df }